I have to agree with On_Headset: the question itself is flawed because it's focusing on the wrong priorities. It should have been phrased more like, "How do you protect the children you're in charge of, the liability of your employing organization, and yourself from a slander law-suit while, hopefully keeping a good atmosphere at the theatre and not embarrasing the man's child?"
Everything from here on out is assuming that I don't live in a "require to report" state. If I did, that's what I'd have to do. I'd let the producer know, first, but I'd report over their objections if necessary, for the good of the kids, me, and the theatre.
Several things I would not do: ask the man straight out what's going on, make any assumptions, whatsoever, or converse about it with anyone who is not in my direct, upward chain of command.
The first thing I would do, when getting these "vague reports" from the boys is to ask for more clarification, on the spot (which I probably wouldn't get, but you have to try.) For all I know, they COULD be uncomfortable with him because he reeks of garlic or forces them to listen to polka music or tales of his latest D&D character's battle with the goblin horde.
Assuming that the reports were made before a show: If I were a male SM, I'd find excuses to pop into the dressing room more frequently until curtain (let me check your makeup because it was a little off last night, wardrobe made a slight adjustment so how's your costume, I heard Bobby had a cold so do you need cough drops, etc.) but as a female SM, I'd have a trusted male crew member/ASM run these little "errands" without letting him know the real reason behind it.
After the show, I'd call the producer and explain what was going on, exactly what the boys said, and any other information my "errands" may have garnered me. I would follow this up in writing, including only the facts and no hypotheses of my own. And I'd insist on a replacement. Tomorrow. I know male volunteers are rare, but they're not one in a million, and we all know an actor who is not currently working, a board member, or a teacher, or a dad with a day job who may not be related to anyone in the cast but can be trusted and can do the job, at least for a few nights. If nothing else, I'd drag my husband in and make him do it until we could get a permenant replacement.
The next day, I would call the volunteer and thank him for his help, but let him know that his services were no longer needed. (Actually, I'd foist this off on the volunteer coordinator, if I could, and in my opinion they would need to be apprised of the facts of the situation.) If he asked why, I'd be honest(ish). "The boys have told me that you're distracting them from focusing on their performance. We've found someone else to watch their room." And for the rest of the conversation, I'd be as nice or as mean as he was, but he would not be coming past the stage door when he dropped his child off for the show, that evening.
